Donnelly, Physics Today, 2009
Pitaevskii & Stringari, BEC, OUP 2003
Second sound is a wave where the normal and superfluid components oscillate with opposite phases.”, Hou, PRA, 2013
An interesting quantity is the Landau-Placzek ratio $\frac{c_p}{c_v} - 1$ which is always small in the case of He.
Taylor, PRA, 2009 and Landau, Fluid Mechanics
For BECs,
Description of the superfluid in terms of normal and superfluid component:
Donnelly, Physics Today, 2009
Linear equations ⇒ look for plane wave solutions.
Plane wave solutions must satisfy a fourth order equation on sound velocity $u$:
\begin{equation} u^4 - \left( \left.\frac{\partial P}{\partial \rho}\right|_{\tilde{s}} + \frac{\rho_s T \tilde{s}^2}{\rho_n \tilde{c}_v}\right) u^2 + \frac{\rho_s T \tilde{s}^2}{\rho_n \tilde{c}_v} \left.\frac{\partial P}{\partial \rho}\right|_{T} = 0 \end{equation}In this regime, we use Bogoliubov thermodynamics. The quasiparticles excitation spectrum is given by:
\[E(\vec{p}) = \sqrt{\frac{p^2}{2m}\left( \frac{p^2}{2m} + 2 g n\right)}\]We express everything in terms of dimensionless units:
An Important parameter to have in mind is: \[\eta = \frac{gn}{k_B T_c} \approx 0.03\]
For example, for the free energy, we get:
\begin{equation} \frac{F(\tilde{t}\,)}{gn N}= E_0(na^3) + \eta^{3/2} \tilde{f}(\tilde{t}) \end{equation}Express the normal density using Landau's formula:
\begin{equation} \rho_n = - \frac{1}{3} \int \frac{\mathrm{d} N_\mathbf{p}(\varepsilon)}{\mathrm{d} \varepsilon} p^2 \frac{\mathrm{d} \mathbf{p}}{(2\pi \hbar)^3} \end{equation}Below hybridization, same behaviour as in liquid helium:
Ideal Bose gas thermodynamics except for \[\frac{1}{\chi_T} = \frac{gn}{m}.\]
For example:
\[\frac{\rho_n}{\rho} = \left(\frac{T}{T_c}\right)^{3/2}\]Yes ! Beliaev theory. (Giorgini, New Journal of Physics, 2010).
Self-consistent equations. For example:
\[n = n_0 + \sum_{\mathbf{k}} \left[ \frac{\varepsilon_k + gn_0}{2E_k} (1+2N_E) - \frac{1}{2}\right]\]where
\[\left\{\begin{array}{l} \varepsilon_k = \hbar^2 k^2 / (2m)\\ E_k = \sqrt{\varepsilon_k (\varepsilon_k + 2gn_0)} \\ N_E = \left(e^{\beta E} - 1\right)^{-1}\\ \end{array}\right.\]
Some problems with the derivatives ($c_v$, $\chi_T$).
Work in progress…
Thanks to S. Stringari, L. P. Pitaevskii, S. Giorgini.